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dc.contributor.authorGasull Embid, Armengol
dc.contributor.authorMañosa Fernández, Víctor
dc.contributor.authorXarles Ribas, Xavier
dc.contributor.otherUniversitat Politècnica de Catalunya. Departament de Matemàtica Aplicada III
dc.date.accessioned2010-05-06T12:34:53Z
dc.date.available2010-05-06T12:34:53Z
dc.date.issued2010-04-30
dc.identifier.urihttp://hdl.handle.net/2117/7135
dc.description.abstractConsider the celebrated Lyness recurrence $x_{n+2}=(a+x_{n+1})/x_{n}$ with $a\in\Q$. First we prove that there exist initial conditions and values of $a$ for which it generates periodic sequences of rational numbers with prime periods $1,2,3,5,6,7,8,9,10$ or $12$ and that these are the only periods that rational sequences $\{x_n\}_n$ can have. It is known that if we restrict our attention to positive rational values of $a$ and positive rational initial conditions the only possible periods are $1,5$ and $9$. Moreover 1-periodic and 5-periodic sequences are easily obtained. We prove that for infinitely many positive values of $a,$ positive 9-period rational sequences occur. This last result is our main contribution and answers an open question left in previous works of Bastien \& Rogalski and Zeeman. We also prove that the level sets of the invariant associated to the Lyness map is a two-parameter family of elliptic curves that is a universal family of the elliptic curves with a point of order $n, n\ge5,$ including $n$ infinity. This fact implies that the Lyness map is a universal normal form for most birrational maps on elliptic curves.
dc.format.extent22 p.
dc.language.isoeng
dc.relation.ispartofseriesarXiv:1004.5511v1 [math.DS]
dc.subjectÀrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals::Equacions en diferències
dc.subject.lcshRecurrent sequences (Mathematics)
dc.subject.otherLyness difference equations
dc.subject.otherRational points over elliptic curves
dc.subject.otherPeriodic points
dc.subject.otherUniversal family of elliptic curves
dc.titleRational periodic sequences for the Lyness recurrence
dc.typeOther
dc.subject.lemacSeqüències recurrents
dc.contributor.groupUniversitat Politècnica de Catalunya. CoDAlab - Control, Modelització, Identificació i Aplicacions
dc.subject.amsClassificació AMS::39 Difference and functional equations::39A Difference equations
dc.subject.amsClassificació AMS::14 Algebraic geometry::14H Curves
dc.relation.publisherversionhttp://arxiv.org/abs/1004.5511
dc.rights.accessOpen Access
local.identifier.drac2316532
dc.description.versionPreprint
local.personalitzacitaciotrue


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